frailtypack (version 2.4.1)

additivePenal: Fit an Additive Frailty model using a semiparametric penalized likelihood estimation or a parametric estimation

Description

Fit an additive frailty model using a semiparametric penalized likelihood estimation or a parametric estimation. The main issue in a meta-analysis study is how to take into account the heterogeneity between trials and between the treatment effects across trials. Additive models are proportional hazard model with two correlated random trial effects that act either multiplicatively on the hazard function or in interaction with the treatment, which allows studying for instance meta-analysis or multicentric datasets. Right-censored data are allowed, but not the left-truncated data. A stratified analysis is possible (maximum number of strata: 2). This approach is different from the shared frailty models. In an additive model, the hazard function for the $j^{th}$ subject in the $i^{th}$ trial with random trial effect $u_i$ as well as the random treatment-by-trial interaction $v_i$ is: $$\left{ \begin{array}{ll} \lambda_{ij}(t|u_i,v_i)=\lambda_0(t)exp(u_i+v_iX_{ij1}+\sum_{k=1}^{p}\beta_kX_{ijk}) \ \bold{cov}(u_i,v_i)=\bold{\rho\sigma\tau} \ u_i\sim\bold{\mathcal{N}}(0,\bold{\sigma^2}) \mbox{,} v_i\sim\bold{\mathcal{N}}(0,\bold{\tau^2}) \end{array} \right.$$ where $\lambda_0(t)$ is the baseline hazard function, $\beta_k$ the fixed effect associated to the covariate $X_{ijk}$ (k=1,..,p), $\beta_1$ is the treatment effect and $X_{ij1}$ the treatment variable. $\rho$ is the corresponding correlation coefficient for the two frailty terms.

Usage

additivePenal(formula, data, correlation = FALSE, recurrentAG =
             FALSE, cross.validation = FALSE, n.knots, kappa1,
             kappa2, maxit = 350, hazard = "Splines", nb.int1)

Arguments

formula

a formula object, with the response on the left of a $\texttildelow$ operator, and the terms on the right. The response must be a survival object as returned by the 'Surv' function like in survival package. The slope() function is requir

data

a 'data.frame' in which to interpret the variables named in the 'formula'.

correlation

Logical value. Are the random effects correlated? If so, the correlation coefficient is estimated. The default is FALSE.

recurrentAG

Always FALSE for additive models (left-truncated data are not allowed).

cross.validation

Logical value. Is cross validation procedure used for estimating smoothing parameter in the penalized likelihood estimation? If so a search of the smoothing parameter using cross validation is done, with kappa1 as the seed.

n.knots

integer giving the number of knots to use. Value required in the penalized likelihood estimation. It corresponds to the (n.knots+2) splines functions for the approximation of the hazard or the survival functions. Number of knots must be between 4

kappa1

positive smoothing parameter in the penalized likelihood estimation. The coefficient kappa of the integral of the squared second derivative of hazard function in the fit. To obtain an initial value for kappa1 (or kappa2), a sol

kappa2

Positive smoothing parameter in the penalized likelihood estimation for the second stratum when data are stratified. See kappa1.

maxit

maximum number of iterations for the Marquardt algorithm. Default is 350

hazard

Type of hazard functions: "Splines" for semiparametric hazard functions with the penalized likelihood estimation, "Piecewise-per" for piecewise constant hazards functions using percentile, "Piecewise-equi" for piecewise constant hazard functions using equ

nb.int1

Number of intervals (between 1 and 20) for the parametric hazard functions ("Piecewise-per", "Piecewise-equi").

Value

• An additive model or more generally an object of class 'additivePenal'. Methods defined for 'additivePenal' objects are provided for print, plot and summary.
• b
  sequence of the corresponding estimation of the splines coefficients, the random effects variances and the regression coefficients.
• call
  The code used for fitting the model.
• coef
  the regression coefficients.
• cov
  covariance between the two frailty terms $(\bold{cov}(u_i,v_i))$
• cross.Val
  Logical value. Is cross validation procedure used for estimating the smoothing parameters in the penalized likelihood estimation?
• correlation
  Logical value. Are the random effects correlated?
• DoF
  degrees of freedom associated with the "kappa".
• formula
  the formula part of the code used for the model.
• groups
  the maximum number of groups used in the fit.
• kappa
  A vector with the smoothing parameters in the penalized likelihood estimation corresponding to each baseline function as components.
• lam
  matrix of hazard estimates and confidence bands.
• lam2
  the same value as lam for the second stratum.
• loglikPenal
  the complete marginal penalized log-likelihood in the semiparametric case.
• loglik
  the marginal log-likelihood in the parametric case.
• n
  the number of observations used in the fit.
• n.events
  the number of events observed in the fit.
• n.iter
  number of iterations needed to converge.
• n.knots
  number of knots for estimating the baseline functions.
• n.strat
  number of stratum.
• rho
  the corresponding correlation coefficient for the two frailty terms.
• sigma2
  Variance for the random intercept (the random effect associated to the baseline hazard functions).
• surv
  matrix of baseline survival estimates and confidence bands.
• surv2
  the same value as surv for the the second stratum.
• tau2
  Variance for the random slope (the random effect associated to the treatment effect across trials).
• varH
  the variance matrix of all parameters before positivity constraint transformation (Sigma2, Tau2, the regression coefficients and the spline coefficients). Then after, the delta method is needed to obtain the estimated variance parameters.
• varHIH
  the robust estimation of the variance matrix of all parameters (Sigma2, Tau2, the regression coefficients and the spline coefficients).
• varSigma2
  The variance of the estimates of "sigma2".
• varTau2
  The variance of the estimates of "tau2".
• varcov
  Variance of the estimates of "cov".
• x1
  vector of times where both survival and hazard functions for the recurrent events are estimated. By default seq(0,max(time),length=99), where time is the vector of survival times.
• x2
  vector of times for the second stratum (see x1 value).
• type.of.hazard
  Type of hazard functions (0:"Splines", "1:Piecewise", "2:Weibull").
• type.of.Piecewise
  Type of Piecewise hazard functions (1:"percentile", 0:"equidistant").
• nbintervR
  Number of intervals (between 1 and 20) for the parametric hazard functions ("Piecewise-per", "Piecewise-equi").
• npar
  number of parameters.
• nvar
  number of explanatory variables.
• noVar
  indicator of explanatory variable.
• LCV
  the approximated likelihood cross-validation criterion in the semiparametric case (with H minus the converged hessien matrix, and l(.) the full log-likelihood).$$LCV=\frac{1}{n}(trace(H^{-1}_{pl}H) - l(.))$$
• AIC
  the Akaike information Criterion for the parametric case.$$AIC=\frac{1}{n}(np - l(.))$$
• n.knots.temp
  initial value for the number of knots.
• shape.weib
  shape parameter for the weibull hazard function.
• scale.weib
  scale parameter for the weibull hazard function.
• martingale.res
  martingale residuals for each cluster.
• frailty.pred
  empirical Bayes prediction of the first frailty term.
• frailty.pred2
  empirical Bayes prediction of the second frailty term.
• linear.pred
  linear predictor: uses simply "Beta'X + u_i + v_i * X_1" in the additive Frailty models.
• global_chisq
  a vector with the values of each multivariate Wald test.
• dof_chisq
  a vector with the degree of freedom for each multivariate Wald test.
• global_chisq.test
  a binary variable equals to 0 when no multivariate Wald is given, 1 otherwise.
• p.global_chisq
  a vector with the p_values for each global multivariate Wald test.
• names.factor
  Names of the "as.factor" variables.
• Xlevels
  vector of the values that factor might have taken.
• contrasts
  type of contrast for factor variable.

Details

The estimated parameter are obtained by maximizing the penalized log-likelihood or by a simple log-likelihood (in the parametric case) using the robust Marquardt algorithm (Marquardt,1963). The parameters are initialized with values obtained with Cox proportional hazard model. The iterations are stopped when the difference between two consecutive loglikelhoods was small $(<10^{-4})$, the="" estimated="" coefficients="" were="" stable="" (consecutive="" values="" $(<10^{-4})$,="" and="" gradient="" small="" enough="" $(<10^{-3})$.="" to="" be="" sure="" of="" having="" a="" positive="" function="" at="" all="" stages="" algorithm,="" spline="" reparametrized="" each="" stage.="" variance="" space="" two="" random="" effects="" is="" reduced,="" so="" variances="" are="" positive,="" correlation="" coefficient="" constrained="" between="" -1="" 1.="" marginal="" log-likelihood="" depends="" on="" integrations="" that="" approximated="" by="" using="" laplace="" integration="" technique="" with="" first="" order="" approximation.="" smoothing="" parameter="" can="" fixed="" or="" maximizing="" likelihood="" cross-validation="" criterion.="" usual="" squared="" wald="" statistic="" was="" modified="" mixture="" $\chi^2$="" distribution="" get="" significance="" test="" for="" effects.="" INITIAL VALUES The splines and the regression coefficients are initialized to 0.1. An adjusted Cox model is fitted, it provides new initial values for the splines coefficients and the regression coefficients. The variances of the frailties are initialized to 0.1. Then an additive frailty model with independent frailties is fitted. At last, an additive frailty model with correlated frailties is fitted.

References

V. Rondeau, Y. Mazroui and J. R. Gonzalez (2012). Frailtypack: An R package for the analysis of correlated survival data with frailty models using penalized likelihood estimation or parametric estimation. Journal of Statistical Software 47, 1-28. V. Rondeau, S. Michiels, B. Liquet, and J. P. Pignon (2008). Investigating trial and treatment heterogeneity in an individual patient data meta-analysis of survival data by mean of the maximum penalized likelihood approach. Statistics in Medecine, 27, 1894-1910.

See Also

slope

Examples

### Additive model with 1 covariate ###

data(dataAdditive)

modAdd <- additivePenal(Surv(t1,t2,event)~cluster(group)+
var1+slope(var1),correlation=TRUE,data=dataAdditive,
n.knots=8,kappa1=10000)

# Var1 is boolean as a treatment variable